In April, 1998, the basketball players Michael Jordan and Shaquille (Shaq) O’Nea
ID: 3006644 • Letter: I
Question
In April, 1998, the basketball players Michael Jordan and Shaquille (Shaq) O’Neal were vying for the season individual scoring title until the last game of the season. The scoring title is won by the player with the highest average number of points per game, calculated by dividing the total number of points by the number of games the player has played. Before the last game, Jordan had scored 2,313 points in 81 games and Shaq had scored 1,666 points in 59 games. No one else had a chance to with the title. Given the above information, with what numbers of points in their final games does Shaq win the scoring title over Jordan? If Jordan scores more points in the last game than Shaq, will he necessarily win? If Shaq scores more points in the last game than Jordan, will he necessarily win?
Explanation / Answer
So the first thing I thought would be useful would be to see what their averages were going into the final game. Jordan’s average would be the 2313 points that he has scored So the first thing I thought would be useful would be to see what their averages were going into the final game. Jordan’s average would be the 2313 points that he has scored
divided by the 81 games that he has played in, which comes out to an average of 28.6 points per game. Shaq’s scoring average would be the 1666 points he has scored divided by the 59 games he has played in, which comes out to an average of 28.2 points per game. So comparing the averages, we see Jordan has a slightly higher average than Shaq, but Shaq has had the “benefit” of having played in fewer games, which means he doesn’t need to score as many points in his last game to raise his average by a certain amount as Jordan would have to score in his last game to raise his average by that same amount. So this question would be almost trivial if Jordan and Shaq had played the same amount of games, but since they didn’t it makes the problem more interesting. The second thing to note is that there are an infinite amount of solutions. Even before doing any specific mathematics we can come to this conclusion. For example, let’s say Jordan plays and he doesn’t score any points, there will be a certain amount of points that Shaq would need to score to raise his average higher than Jordan’s. Let’s say this amount of points is p0. So if Jordan does score zero points his last game, Shaq would have to score at least p0 points, or in other words the number of points Shaq scoresp0, which constitutes an infinite solution. Realistically, a basketball player will probably not score 100 points or more in a game, so we can cap off the number of points a player can score in a game, at least for the sake of coming up with a non-infinite solution to the problem.
So the first thing I thought would be helpful was to make variables and set up equations. So I let x=the number of points Shaq scores in the last game and I let y=the number of points Jordan scores in the last game. With these two variables we can set up an equation:
(1666+x)/60=(2313+y)/82
This equation represents the fact that Jordan and Shaq’s scoring average will be the same after the last game, with Shaq’s average equation on the left and Jordan’s average equation on the right. With this equation we can come up with different representations of the solutions.
The first thing, that I thought might be interesting to find was if both players scored the same number of points in the last game, what number of points would give them the same scoring average after the last game. To do this, I let x=y, since we are assuming both players scored the same amount of points, so our equation becomes:
(1666+x)/60=(2313+x)/82
With this, we can just cross multiply, which you can do because it is the same thing as multiplying each fraction by 1 in the form n/n, where n is the denominator of the opposite fraction. When we do this and distribute through, we get 138780+60x=136612+82x. Next, I just subtracted 60x and 136612 from both sides, to preserve the equality of the statement, sides to get 2168=22x. Next, I divided both sides by 22 because it preserves the equality of the statement and because dividing by 22 is the inverse operation of multiplying by 22 which we need to do to get the x by itself on one side. When we do this, we end with x=98.545454. So this is the number of points that both players would have to score to have the same scoring average, assuming they score the same amount of points. Note that this situation is in fact impossible since the solution is not an integer solution. It is impossible for a player to score a fraction of a point, so it turned out that this solution did not really particularly help me.
Moving to a numerical representation, we can take a case by case scenario where we fix the amount of points Jordan scores and calculate the least number of points Shaq would need
to score in order to obtain a higher scoring average than Jordan. So I started with the case when Jordan scores zero points in the last game. If this happens then Jordan’s scoring average after the final game will be (2313+0)/82=2313/82. The question of the problem asks how many points Shaq and Jordan would need to score in order for Shaq to win the scoring title. So if Jordan scores zero points his last game, we are looking for when Shaq’s average>2313/82 (Jordan’s average). Plugging in the equation for Shaq’s average, we get (1666+x)/60>2313/82. We would then cross multiply and get 136612+82x>138780. We then subtract 136612 from both sides, to preserve the equality of the statement, and we end up with 82x>2168. Dividing by 82 on both sides, we are left with x>26.439…… We are only concerned with the integer solutions, because basketball players can not score fractions of a point, so the solution will turn out to be x=27,28,29,……,144 (note: 144 is going to be our cap for the amount of points a player scores in a game, since that is the most points a player has ever scored in a regulation game). If Jordan scores 1 point in the last game, his scoring average would be (2313+1)/82=2314/82. So we are looking for the solution to the following inequality: (1666+x)/60>2314/82. Using the same procedure as we used when Jordan scored zero points, we end up getting x>27.17…… But since we are only concerned with integer solutions, the solution will turn out to be x=28,29,…….,144, which represents the number of points Shaq would need to score if Jordan scored 1 point. We can continue this process and we get the following results:
When Jordan scores Shaq would need to score
0 points 27,28,29,…….,144 points
1 point 28,29,30,……,144 points
2 points 28,29,30,……...,144 points
3 points 29,30,31,………..,144 points
4 points 30,31,32,…………,144 points
5 points 31,32,33,………….,144 points
We could keep going with this procedure until we get to the case where Jordan scores 144 points, our cap for this problem, but that would obviously be very tedious. While doing this, I knew there had to be an easier way to figure this out, so I thought trying to come up with an algebraic representation would be good to tackle next.
Coming up with the algebraic representation was a little bit tricky, although the representation was, in essence, used for the numerical approach. I had trouble getting around the issue of dependency and I ended up getting an equation that had the number of points that Shaq needed to score his last game depending on the number of points Jordan scored in his last game. In a way that makes sense, because we are dealing with averages and obviously the number of points of Shaq scores in his last game will depend on the number of points Jordan scores, and of course the problem wants us to find when Shaq would win the scoring title. I am still not 100% sure if that is the correct reasoning but I went with it anyway. So we have Shaq and Jordan’s scoring average after the last game, which are (1666+x)/60 and (2313+y)/82 respectively, and we just set Shaq’s average greater than Jordan’s average and we get:
(1666+x)/60>(2313+y)/82
From this we need to solve for x so we get an inequality that depends on the number of points Jordan scored. So we cross multiply and distribute to get 136612+82x>138780+60y. From here we subtract 136612 from both sides to preserve the equality of the statement and we get 82x>60y+2168. We then divide both sides by 82 and we end up with the inequality x>(60y+2168)/82. Like I said, although I didn’t realize it at the time, this is pretty much the inequality I used to come up with the numerical representation. Now with this algebraic representation we can easily obtain the number of points Shaq would need, given the amount of points Jordan scores, in order to obtain a higher scoring average than Jordan. With this algebraic representation, we can now easily obtain a graphical representation of the solution.
All we have to do to obtain the graphical representation of the solution is take the algebraic representation and graph what the inequality would look like. The first thing we should note is that the inequality we will be graphing is a line, since the highest power of y is 1. To graph this we just start at the intercept, which in this case is 2168/82 and use the slope, which in this case is 60/82, to extend it. Secondly, the line we will be graphing will in fact be a dotted line, since we are graphing an inequality that involves greater than. Since the inequality is a greater than type, we would normally just shade above the line, but since the context of the problem only allows for integer solutions, in fact just positive integer solutions, I showed this by dotting the individual integer solutions above the dotted line. We also eliminate all the solutions involving negative numbers since it is impossible for a basketball player to score a negative number of points.